Search: id:A000930 Results 1-1 of 1 results found. %I A000930 M0571 N0207 %S A000930 1,1,1,2,3,4,6,9,13,19,28,41,60,88,129,189,277,406, %T A000930 595,872,1278,1873,2745,4023,5896,8641,12664,18560,27201, %U A000930 39865,58425,85626,125491,183916,269542,395033,578949,848491 %N A000930 a(n) = a(n-1) + a(n-3). %C A000930 Might be called the Middle Sequence, since it is a cross between the Fibonacci sequence A000045 and the Padovan sequence A000931. - Kyle Ledbetter Lucas, Jun 13 2009 %C A000930 A Lam{\'e} sequence of higher order. %C A000930 Could have begun 1,0,0,1,1,1,2,3,4,6,9,... but that would spoil many nice properties. %C A000930 Number of tilings of a 3 X n rectangle with straight trominoes. %C A000930 Number of ways to arrange n-1 tatami mats in a 2 X (n-1) room such that no 4 meet at a point. For example, there are 6 ways to cover a 2 X 5 room, described by 11111, 2111, 1211, 1121, 1112, 212. %C A000930 Equivalently, number of ordered partitions of n-1 into parts 1 and 2 with no two 2's adjacent. E.g. there are 6 such ways to partition 5, namely 11111, 2111, 1211, 1121, 1112, 212, so a(9) = 6. %C A000930 This comment covers a family of sequences which satisfy a recurrence of the form a(n) = a(n-1) + a(n-m), with a(n) = 1 for n = 0...m-1. The generating function is 1/(1-x-x^m). Also a(n) = sum(binomial(n-(m-1)*i, i), i=0..n/m). This family of binomial summations or recurrences gives the number of ways to cover (without overlapping) a linear lattice of n sites with molecules that m sites wide. Special case: m=1: A000079; m=4: A003269; m=5: A003520; m=6: A005708; m=7: A005709; m=8: A005710. %C A000930 a(n-4) = number of n-bit sequences that start and end with 0 but avoid both 010 and 0110. - David Callan (callan(AT)stat.wisc.edu), Mar 25 2004 %C A000930 a(n+2) = number of n-bit 0-1 sequences that avoid both 00 and 010. - David Callan (callan(AT)stat.wisc.edu), Mar 25 2004 %C A000930 Also number of compositions of n+1 into parts congruent to 1 mod m. - Vladeta Jovovic (vladeta(AT)eunet.rs), Feb 09 2005 %C A000930 Row sums of Riordan array (1/(1-x^3), x/(1-x^3)). - Paul Barry (pbarry(AT)wit.ie), Feb 25 2005 %C A000930 Row sums of Riordan array (1,x(1+x^2)). - Paul Barry (pbarry(AT)wit.ie), Jan 12 2006 %C A000930 Starting with offset 1 = row sums of triangle A145580 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 13 2008] %C A000930 Number of digits in A061582. [From Dmitry Kamenetsky (dkamen(AT)rsise.anu.edu.au), Jan 17 2009] %D A000930 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 78,80. %D A000930 E. Di Cera and Y. Kong, Theory of multivalent binding in one and two-dimensional lattices, Biophysical Chemistry, Vol. 61 (1996), pp. 107-124. %D A000930 M. Feinberg, New slants, Fib. Quart. 2 (1964), 223-227. %D A000930 T. M. Green, Recurrent sequences and Pascal's triangle, Math. Mag., 41 (1968), 13-21. %D A000930 J. Hermes, Anzahl der Zerlegungen einer ganzen rationalen Zahl in Summanden, Math. Ann., 45 (1894), 371-380. %D A000930 D. Jarden, Recurring Sequences. Riveon Lematematika, Jerusalem, 1966. %D A000930 H. Langman, Play Mathematics. Hafner, NY, 1962, p. 13. %D A000930 Antonio M. Oller-Marc\'en, "The Dying Rabbit Problem Revisited", INTEGERS, 9 (2009), 129-138 [From Parthasarathy Nambi (PachaNambi(AT)yahoo.com), May 09 2009] %D A000930 David Sankoff and Lani Haque, Power Boosts for Cluster Tests, in Comparative Genomics, Lecture Notes in Computer Science, Volume 3678/2005, Springer-Verlag. [Added by N. J. A. Sloane, Jul 09 2009] %D A000930 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). %D A000930 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). %H A000930 T. D. Noe, Table of n, a(n) for n=0..500 %H A000930 J.-P. Allouche and T. Johnson, Narayana's Cows and Delayed Morphisms %H A000930 J.-P. Allouche and T. Johnson, Narayana's Cows and Delayed Morphisms %H A000930 P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 373 %H A000930 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 14 %H A000930 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 376 %H A000930 S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992. %H A000930 S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992. %H A000930 T. Sillke, The binary form of Conway's Sequence %H A000930 E. Wilson, The Scales of Mt. Meru %H A000930 Index entries for sequences related to linear recurrences with constant coefficients %F A000930 G.f.: 1/(1-x-x^3). %F A000930 a(n) = sum(binomial(n-2*i, i), i=0..n/3). %F A000930 a(n) = a(n-2) + a(n-3) + a(n-4) for n>3. %F A000930 a(n) = floor( d*c^n + 1/2) where c is the real root of x^3-x^2-1 and d is the real root of 31*x^3-31*x^2+9*x-1 ( c=1.465571231876768... and d= 0.611491991950812...) - Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 30 2002 %F A000930 a(n)=sum{k=0..n, binomial(floor((n+2k-2)/3), k)}. - Paul Barry (pbarry(AT)wit.ie), Jul 06 2004 %F A000930 a(n)=sum{k=0..n, C(k, floor((n-k)/2))(1+(-1)^(n-k))/2}; - Paul Barry (pbarry(AT)wit.ie), Jan 12 2006 %F A000930 a(n)=sum{k=0..n, C((n+2k)/3,(n-k)/3)*(2*cos(2*pi*(n-k)/3)+1)/3}; - Paul Barry (pbarry(AT)wit.ie), Dec 15 2006 %F A000930 a(n) = term (1,1) in matrix [1,1,0; 0,0,1; 1,0,0]^n - Alois P. Heinz (heinz(AT)hs-heilbronn.de), Jun 20 2008 %F A000930 Contribution from Paul D. Hanna (pauldhanna(AT)juno.com), Oct 08 2009: (Start) %F A000930 G.f.: exp( Sum_{n>=1} [(1+sqrt(1+4x))^n + (1-sqrt(1+4x))^n]*(x/2)^n/n ). %F A000930 Logarithmic derivative equals A001609. (End) %e A000930 sage: taylor( mul(x/(1 - x - x^3) for i in xrange(1,2)),x,0,38)#>> solution: x + x^2 + x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 6*x^7 + 9*x^8 + 13*x^9 + 19*x^10 + 28*x^11 + 41*x^12 + 60*x^13 + 88*x^14 + 129*x^15 + 189*x^16 + 277*x^17 + 406*x^18 + 595*x^19 + 872*x^20 + 1278*x^21 + 1873*x^22 + 2745*x^23 + 4023*x^24 + 5896*x^25 + 8641*x^26 + 12664*x^27 + 18560*x^28 + 27201*x^29 + 39865*x^30 + 58425*x^31 + 85626*x^32 + 125491*x^33 + 183916*x^34 + 269542*x^35 + 395033*x^36 + 578949*x^37 + 848491*x^38 [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 29 2009] %p A000930 f := proc(r) local t1,i; t1 := []; for i from 1 to r do t1 := [op(t1), 0]; od: for i from 1 to r+1 do t1 := [op(t1),1]; od: for i from 2*r+2 to 50 do t1 := [op(t1),t1[i-1]+t1[i-1-r]]; od: t1; end; # set r = order %p A000930 with(combstruct): SeqSetU := [S, {S=Sequence(U), U=Set(Z, card > 2)}, unlabeled]: seq(count(SeqSetU, size=j), j=3..40); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 10 2006 %p A000930 seq(add(binomial(n-2*k,k),k=0..floor(n/3)),n=0..37); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 03 2007 %p A000930 BB :=-x/(-1+x^2+x^6); series(BB,x,77); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 24 2007 %p A000930 A000930:=-1/(-1+z+z**3); [S. Plouffe in his 1992 dissertation.] %p A000930 a := n -> (Matrix([[1,1,0],[0,0,1],[1,0,0]])^n)[1,1]; seq ((a(n)), n=0..50); - Alois P. Heinz (heinz(AT)hs-heilbronn.de), Jun 20 2008 %t A000930 a[0] = 1; a[1] = a[2] = 1; a[n_] := a[n] = a[n - 1] + a[n - 3]; Table[ a[n], {n, 0, 40} ] %t A000930 CoefficientList[Series[1/(1-x-x^3), {x, 0, 45}], x] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 22 2007 %o A000930 (Other) sage: taylor( mul(x/(1 - x - x^3) for i in xrange(1,2)),x,0,38)# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 29 2009] %o A000930 (PARI) {a(n)=polcoeff(exp(sum(m=1,n,((1+sqrt(1+4*x))^m + (1-sqrt(1+4*x))^m)*(x/ 2)^m/m)+x*O(x^n)),n)} [From Paul D. Hanna (pauldhanna(AT)juno.com), Oct 08 2009] %Y A000930 For Lam{\'e} sequences of orders 1 through 9 see A000045, this one, A017898-A017904. %Y A000930 Cf. A048715, A069241. %Y A000930 See also A000079, A003269, A003520, A005708, A005709, A005710. %Y A000930 Essentially the same as A068921 and A078012. %Y A000930 A145580 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 13 2008] %Y A000930 Cf. A001609. [From Paul D. Hanna (pauldhanna(AT)juno.com), Oct 08 2009] %Y A000930 Sequence in context: A017825 A159848 A017826 this_sequence A068921 A078012 A135851 %Y A000930 Adjacent sequences: A000927 A000928 A000929 this_sequence A000931 A000932 A000933 %K A000930 nonn,easy,nice %O A000930 0,4 %A A000930 N. J. A. Sloane (njas(AT)research.att.com). Search completed in 0.002 seconds